Projective classification of rational \(\mathbb {C}\mathbf {P}^{1}\)-mappings

Abstract

We study the orbits of various \(\mathbf {SL}_{2}\left( \mathbb {C}\right) \)-actions on the spaces of rational \(\mathbb {C}\mathbf {P}^{1}\)-mappings. The fields of rational differential invariants and the corresponding ordinary differential equations that describe orbits are found.

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References

  1. 1.

    Bibikov, P., Lychagin, V.: Projective classification of binary and ternary forms. J. Geom. Phys. 61(10), 1914–1927 (2011)

    Google Scholar 

  2. 2.

    Bibikov, P., Lychagin, V.: On differential invariants of actions of semisimple Lie groups. J. Geom. Phys. 85, 99–105 (2014)

    Google Scholar 

  3. 3.

    Bogomolov, F., Petrov, T.: Algebraic curves and one-dimensional fields. Courant Lect. Math. 8, 214 (2002)

    Google Scholar 

  4. 4.

    Konovenko, N., Lychagin, V.: On projective classification of algebraic curves. Math. Bull. Shevchenko Sci. Soc. 10, 51–64 (2013). https://doi.org/10.1007/s13324-015-0113-5

    Google Scholar 

  5. 5.

    Krasilshchik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of jet spaces and nonlinear partial differential equations. Advanced Studies in Contemporary Mathematics, vol. 1. Gordon and Breach Science Publishers, New York, xx+441 pp (1986)

  6. 6.

    Kruglikov, B., Lychagin, V.: Geometry of differential equations. In: Hanbook of Global Analysis, pp. 725–772 (2008)

  7. 7.

    Konovenko, N.: Differential Invariants and \(\mathfrak{sl}_{2}\)-Geometries, p. 188. Naukova Dumka, Kiev (2013). (in Russian)

    Google Scholar 

  8. 8.

    Kumpera, A.: Invariants differentiels d’un pseudogroupe de Lie, Part 1. J. Differ. Geom. 10(2), 289–345 (1975)

    Google Scholar 

  9. 9.

    Kumpera, A.: Invariants differentiels d’un pseudogroupe de Lie, Part 2. J. Differ. Geom. 10(3), 347–416 (1975)

    Google Scholar 

  10. 10.

    Kruglikov, B., Lychagin, V.: Global Lie–Tresse theorem. Sel. Math. 22(3), 1357–1411 (2016)

    Google Scholar 

  11. 11.

    Olver, P.: Classical Invariant Theory. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  12. 12.

    Olver, P.: Differential invariant algebras. Comtemp. Math. 549, 95–121 (2011)

    Google Scholar 

  13. 13.

    Ovsiannikov, L.: Group Analysis of Differential Equations, pp. 487–489. Nauka, Moscow (1978). (Engl. transl. Academic Press New York (1982))

    Google Scholar 

  14. 14.

    Rosenlicht, M.: A remark on quotient spaces. An. Acad. Brasil. Ci. 35, 487–489 (1963)

    Google Scholar 

  15. 15.

    Tresse, A.: Sur les invariants differentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)

    Google Scholar 

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Correspondence to Konovenko Nadiia.

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Nadiia, K., Valentin, L. Projective classification of rational \(\mathbb {C}\mathbf {P}^{1}\)-mappings. Anal.Math.Phys. 9, 1865–1876 (2019). https://doi.org/10.1007/s13324-019-00281-2

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